Sheaf cohomology as the derived functor of global sections (Tag D-COHDER)
For \(X\) a topological space, \(\globsec{X; \wait}: \Sh(X; \Ab) \to \Ab\) is left exact. Define \(H^i(X, \mcf) \da R^i \globsec{X; \wait}(\mcf)\): choose an injective resolution \(\mcf \injects \mci^\bullet\), apply \(\globsec{X; \wait}\), and take cohomology.
(Enough injectives)
The construction requires enough injective objects. For each \(x\), embed the stalk \(\mcf_x \injects I_x\) into an injective \(\OO_{X,x}\dash\)module, let \(j^x: \ts{x} \injects X\), and set \(\mci \da \prod_{x \in X} j^x_* I_x\). Then \(\Hom(\mcg, \mci) = \prod_x \Hom_{\OO_{X,x}}(\mcg_x, I_x)\), a composite of the exact stalk functors with exact \(\Hom\) functors, so \(\mci\) is injective, and \(\mcf \injects \mci\).
(Acyclic resolutions)
For every resolution \(\mcf\to\mcl^\bullet\) by \(\globsec{X;\wait}\dash\)acyclic sheaves, \(H^i(X,\mcf)\cong H^i(\globsec{X;\mcl^\bullet})\) [@Har10a, Proposition III.1.2A]. Flasque sheaves are acyclic [@Har10a, Proposition III.2.5], and every sheaf embeds in the flasque sheaf \(U\mapsto\prod_{x\in U}\mcf_x\) of discontinuous sections. Hence \(H^i\) is effaceable for \(i>0\), and \((H^i)_{i\ge0}\) is a universal \(\delta\dash\)functor [@Har10a, Theorem III.1.3A]: for every \(\delta\dash\)functor \((T^i)\) from \(\Sh(X;\Ab)\) to \(\Ab\), every natural transformation \(H^0\to T^0\) extends uniquely to a morphism of \(\delta\dash\)functors \((H^i)\to(T^i)\).
Any resolution of \(\mcf\) by \(\globsec{X;\wait}\)-acyclic sheaves computes \(H^*(X,\mcf)\).
Flasque sheaves and acyclicity (Tag D-COHFLQ)
\(\mcf\) is flasque, or flabby, if every restriction \(\mcf(U) \to \mcf(V)\), for \(V \subseteq U\), is surjective.
Injective implies flasque, flasque implies \(\globsec{X;\wait}\dash\)acyclic, and for a short exact sequence with flasque left term the sequence of global sections stays exact.
Every sheaf \(\mcf\) embeds in the flasque sheaf \(\prod_x j^x_* \mcf_x\) of discontinuous sections, so every sheaf has a resolution by flasque, hence \(\globsec{X;\wait}\dash\)acyclic, sheaves.
The proof of acyclicity is induction on the sequence \(0 \to \mcf \to \mci \to \mcg \to 0\) with \(\mci\) injective: flasqueness of \(\mcf\) forces \(\mcg\) flasque and the global sections exact, so \(H^1(\mcf) = 0\) and the higher groups shift down.
(Flasque sheaves)
The following sheaves are flasque:
the sheaf \(\prod_x j^x_* \mcf_x\) of discontinuous sections of any sheaf \(\mcf\), where \(j^x\colon\ts{x}\injects X\);
every constant sheaf on an irreducible topological space [@Har10a, Exercise II.1.16];
the sheaf \(\tilde{I}\) on \(\Spec A\) for a Noetherian ring \(A\) and an injective \(A\dash\)module \(I\) [@Har10a, Proposition III.3.4]; Hartshorne uses this to prove that \(H^i(\Spec A,\tilde M)=0\) for \(i>0\).
Since flasque sheaves are \(\globsec{X;\wait}\)-acyclic, a flasque resolution, such as the Godement resolution, computes sheaf cohomology.
Soft and fine sheaves (Tag D-SHFFINE)
Let \(X\) be a paracompact Hausdorff space and \(\mathcal{F}\) a sheaf of abelian groups on \(X\).
\(\mathcal{F}\) is soft if for every closed \(Z \subseteq X\) the restriction \(\Gamma(X, \mathcal{F}) \to \Gamma(Z, \mathcal{F}|_Z)\) is surjective.
\(\mathcal{F}\) is c-soft if for every compact \(K \subseteq X\) the restriction \(\Gamma(X, \mathcal{F}) \to \Gamma(K, \mathcal{F}|_K)\) is surjective.
\(\mathcal{F}\) is fine if for every locally finite open cover \(\{U_i\}\) of \(X\) there are sheaf endomorphisms \(\eta_i \colon \mathcal{F} \to \mathcal{F}\) with the support of \(\eta_i\) contained in \(U_i\) and \(\sum_i \eta_i = \mathrm{id}_{\mathcal{F}}\).
On a paracompact Hausdorff space, flasque sheaves and fine sheaves are soft, and soft sheaves are acyclic: \(H^i(X, \mathcal{F}) = 0\) for \(i > 0\). So a resolution of a sheaf by fine sheaves computes its cohomology.
On a smooth manifold \(M\), the sheaf \(\mathcal{C}^\infty_M\) of smooth functions is fine, using a smooth partition of unity subordinate to the cover, and so is every sheaf of \(\mathcal{C}^\infty_M\)-modules, such as the sheaves \(\mathcal{A}^p\) of smooth \(p\)-forms. The de Rham complex \(0 \to \ul{\RR} \to \mathcal{A}^0 \to \mathcal{A}^1 \to \cdots\) is a fine resolution of the constant sheaf, which proves \(H^p(M, \RR) \cong H^p_{\mathrm{dR}}(M)\). The sheaf of holomorphic functions on \(\CC\) is not soft: a function holomorphic near the closed set \(\{0, 1\}\), equal to \(0\) near \(0\) and \(1\) near \(1\), does not extend to an entire function.
The Godement resolution (Tag D-COHGODEMENT)
Let \(\mcf\) be a sheaf of abelian groups on a topological space \(X\). Put \[ \mathcal{G}^0(\mcf)(U) = \prod_{x \in U} \mcf_x , \] the sheaf of all, not necessarily continuous, sections of the espace étalé, with the injection \(\mcf \to \mathcal{G}^0(\mcf)\) sending \(s\) to its germs. Inductively, with \(\mathcal{Z}^0 = \mcf\) and \(\mathcal{Z}^{n+1} = \operatorname{coker}\big(\mathcal{Z}^{n} \to \mathcal{G}^0(\mathcal{Z}^{n})\big)\), set \(\mathcal{G}^n(\mcf) = \mathcal{G}^0(\mathcal{Z}^n)\) with differential \(\mathcal{G}^n(\mcf) \to \mathcal{Z}^{n+1} \to \mathcal{G}^{n+1}(\mcf)\). The complex \(0 \to \mcf \to \mathcal{G}^0(\mcf) \to \mathcal{G}^1(\mcf) \to \cdots\) is the Godement resolution.
Each \(\mathcal{G}^n(\mcf)\) is flasque, since a product of stalks over a smaller open set extends by zero to a larger one.
The Godement resolution is exact and functorial in \(\mcf\), and exact as a functor of \(\mcf\), because it is built from stalks.
Flasque sheaves are acyclic, so \(H^i(X, \mcf) = H^i\big(\Gamma(X, \mathcal{G}^\bullet(\mcf))\big)\).
For a continuous map \(f\colon X\to Y\), pushforwards of flasque sheaves are flasque and flasque sheaves are \(f_*\)-acyclic, so \(R^if_*\mcf\cong\mathcal H^i\big(f_*\mathcal G^\bullet(\mcf)\big)\). The complexes \(\Gamma(X,\mathcal G^\bullet(\mcf))\) and \(f_*\mathcal G^\bullet(\mcf)\) are functorial in \(\mcf\) and involve no choice of injective resolution.
Affine and Grothendieck vanishing
Higher cohomology vanishes on an affine scheme (Tag T-COHAFF)
Let \(X = \Spec A\) with \(A\) Noetherian and \(\mcf \in \QCoh(X)\). Then \(H^0(X, \mcf) = \globsec{X;\mcf}\) and \(H^i(X, \mcf) = 0\) for all \(i > 0\).
(The proof in one line)
Put \(M = \globsec{X;\mcf}\), so \(\mcf \cong \tilde M\). Take an injective resolution \(M \injects I^\bullet\) in \(\amod\) and sheafify: each \(\tilde{I^j}\) is flasque, hence acyclic, so \(\tilde{I^\bullet}\) computes the cohomology. Applying \(\globsec{X;\wait}\) returns the original exact complex \(M \injects I^\bullet\), and its higher cohomology is zero.
(Čech computation)
Let \(\mcf\) be a quasicoherent sheaf on a scheme \(X\).
If \(X\) is affine and \(\mathfrak{U} = \{U_1, \ldots, U_r\}\) is a finite cover of \(X\) by affine opens whose finite intersections are affine, the augmented Čech complex \(0 \to \mcf(X) \to C^0(\mathfrak{U}, \mcf) \to C^1(\mathfrak{U}, \mcf) \to \cdots\) is exact; so \(\check{H}^i(\mathfrak{U}, \mcf) = 0\) for \(i > 0\).
If \(X\) is quasicompact and separated, any two finite affine covers \(\mathfrak{U}\), \(\mathfrak{V}\) give the same Čech cohomology: \(\check{H}^i(\mathfrak{U}, \mcf) \cong \check{H}^i(\mathfrak{V}, \mcf)\); for \(X\) Noetherian both equal \(H^i(X, \mcf)\) [@Har10a, Theorem III.4.5].
(Hypotheses)
Quasicoherence is the load-bearing hypothesis, and it is where the whole argument lives: the theorem is false for general sheaves of abelian groups, which see the topology of \(X\) rather than the module \(M\). Noetherianness is only used to know that \(\tilde I\) is flasque for \(I\) injective; the statement is true for any affine scheme, but that version is harder.
This is the direction of Serre’s criterion that is a theorem about affines, and the other direction is the criterion proper. It is also the reason every computation is done on an affine cover: on the pieces of the cover there is nothing above degree \(0\), so all the cohomology comes from the gluing, which is exactly what the Čech complex records.
Grothendieck vanishing above the dimension (Tag T-COHGROTH)
Let \(X\) be a Noetherian topological space of dimension \(n\). Then \(H^i(X, \mcf) = 0\) for all \(i > n\) and every sheaf of abelian groups \(\mcf\) on \(X\).
Two features distinguish this from every other vanishing theorem on the syllabus. It asks nothing of \(\mcf\): no quasicoherence, no coherence. And the bound is topological, the dimension of the underlying space, not the dimension of a variety over a field.
Noetherianness of the space is load-bearing and is used to run the Noetherian induction on closed subsets; without it the statement fails. Nothing else can be dropped, since \(H^n\) is generally nonzero: \(H^n(\PP^n, \OO(-n-1)) \neq 0\) shows the bound is sharp.
For quasicoherent sheaves there is a cheaper route to the same bound on \(\PP^n\): a cover by \(n+1\) affines makes the Čech complex vanish above degree \(n\). The two arguments give the same number for different reasons, and the Čech one says nothing about non-quasicoherent sheaves.
For a quasicoherent sheaf \(\mcf\) on a noetherian separated scheme \(X\) with a finite affine open cover \(\mathfrak U\), every finite intersection of members of \(\mathfrak U\) is affine, affine vanishing makes \(\mcf\) acyclic on each of them, and so the Čech complex of \(\mathfrak U\) computes \(H^*(X,\mcf)\) (Čech cohomology and the comparison with sheaf cohomology). Grothendieck vanishing holds for every sheaf of abelian groups on a noetherian topological space, with bound \(\dim X\). Affine vanishing fails for sheaves that are not quasicoherent: on \(X=\AA^1_k\) with distinct closed points \(P,Q\), \(U=X\setminus\{P,Q\}\), and \(j\colon U\hookrightarrow X\), the sequence \(0\to j_!\ZZ_U\to\ZZ_X\to\ZZ_P\oplus\ZZ_Q\to0\), with skyscraper sheaves \(\ZZ_P,\ZZ_Q\), gives \(H^1(X,j_!\ZZ_U)\cong\ZZ^2/\ZZ\neq0\), because \(\ZZ_X\) is flasque on the irreducible space \(X\) [Hart10, Exercise III.2.1].
The long exact sequence
The long exact sequence, and the ideal sequence as a computational device (Tag PR-COHLES)
A short exact sequence \(0 \to \mcf' \to \mcf \to \mcf'' \to 0\) of sheaves of abelian groups gives a long exact sequence \[ 0 \to H^0(\mcf') \to H^0(\mcf) \to H^0(\mcf'') \mapsvia{\delta} H^1(\mcf') \to \cdots , \] natural in morphisms of short exact sequences.
(Long exact sequence of a short exact sequence of complexes)
Let \(0 \to C'^\bullet \xrightarrow{u} C^\bullet \xrightarrow{v} C''^\bullet \to 0\) be a short exact sequence of complexes in an abelian category. Then there are connecting morphisms \(\delta^i \colon h^i(C''^\bullet) \to h^{i+1}(C'^\bullet)\), natural in the sequence, making \[\cdots \to h^i(C'^\bullet) \to h^i(C^\bullet) \to h^i(C''^\bullet) \xrightarrow{\delta^i} h^{i+1}(C'^\bullet) \to \cdots\] exact.
For derived-functor cohomology, the proposition follows by applying the lemma to \(0 \to \Gamma(X, I'^\bullet) \to \Gamma(X, I^\bullet) \to \Gamma(X, I''^\bullet) \to 0\) for compatible injective resolutions, which remains exact because \(I'^\bullet\) is injective in each degree. For Čech cohomology of quasicoherent sheaves on a quasicompact separated scheme with a finite affine cover \(\mathfrak{U}\), the sequence \(0 \to C^\bullet(\mathfrak{U}, \mcf') \to C^\bullet(\mathfrak{U}, \mcf) \to C^\bullet(\mathfrak{U}, \mcf'') \to 0\) is exact because every finite intersection of the cover is affine and sections of quasicoherent sheaves over affines form exact sequences, and the lemma applies directly.
(What the connecting map is)
\(\delta\) sends a global section of \(\mcf''\) to the obstruction to lifting it: lift locally on a cover, take the differences of the lifts on overlaps, and read the result as a class in \(H^1(\mcf')\). So \(H^1(\mcf') = 0\) is precisely the statement that every global section of the quotient lifts.
(Standard computations)
For \(Y \subseteq \PP^n\) closed with ideal sheaf \(\mci_Y\), twist the ideal sequence: \[ 0 \to \mci_Y(d) \to \OO_{\PP^n}(d) \to \OO_Y(d) \to 0 . \] Known cohomology of the middle term plus known cohomology of the left term determines the right, which is the unknown. For a smooth hypersurface of degree \(e\), \(\mci_Y = \OO(-e)\) and everything is a binomial coefficient; this is how one gets \(h^0(\OO_Y(d))\), the Hilbert polynomial of \(Y\), and the genus of a plane curve as \(\binom{e-1}{2}\).
The other constant use is the skyscraper sequence \(0 \to \OO(D-p) \to \OO(D) \to k(p) \to 0\) on a curve, which is how \(\ell(D)\) moves by at most one when a point is added, and hence how Riemann–Roch is proved by induction.
For a closed subscheme \(Y\subseteq\PP^n\) with ideal sheaf \(\mci_Y\), the twisted ideal sequence \(0\to\mci_Y(d)\to\OO_{\PP^n}(d)\to\OO_Y(d)\to0\) and \(H^i(\PP^n,\OO(d))=0\) for \(0<i<n\) give \(H^i(Y,\OO_Y(d))\cong H^{i+1}(\PP^n,\mci_Y(d))\) for \(1\le i\le n-2\). For a \(k\)-rational point \(p\) on a curve over \(k\), the skyscraper sequence \(0\to\OO(D-p)\to\OO(D)\to k(p)\to0\) gives \(\ell(D)-1\le\ell(D-p)\le\ell(D)\).
The six operations
The six operations on sheaves (Tag D-SHFSIX)
Let \(j \colon U \hookrightarrow X\) be the inclusion of an open subset and \(\mathcal{F}\) a sheaf of abelian groups on \(U\). The extension by zero \(j_! \mathcal{F}\) is the sheafification of the presheaf \(V \mapsto \mathcal{F}(V)\) if \(V \subseteq U\), and \(V \mapsto 0\) otherwise. Its stalks are \((j_! \mathcal{F})_x = \mathcal{F}_x\) for \(x \in U\) and \(0\) for \(x \notin U\); \(j_!\) is exact and left adjoint to \(j^{-1}\).
(Six operations)
For a continuous map \(f \colon X \to Y\) of locally compact Hausdorff spaces, the six operations on bounded-below derived categories of sheaves of abelian groups are \[ f^{-1} \dashv Rf_*, \qquad Rf_! \dashv f^!, \qquad \otimes^L \dashv R\mathcal{H}om . \] Here \(f_!\) is direct image with proper supports: \((f_! \mathcal{F})(V)\) consists of the sections \(s \in \mathcal{F}(f^{-1}V)\) whose support is proper over \(V\). The exceptional inverse image \(f^!\) is the right adjoint of \(Rf_!\); it exists on derived categories when \(f_!\) has finite cohomological dimension, and is not the derived functor of a functor on sheaves.
For \(f\) proper, \(Rf_! = Rf_*\); for \(j\) an open inclusion, \(j^! = j^{-1}\).
Base change: for a Cartesian square with \(g \colon Y' \to Y\) and \(f' \colon X' \to Y'\), \(g^{-1} Rf_! \cong Rf'_! g'^{-1}\).
Projection formula: \(Rf_!(\mathcal{F} \otimes^L f^{-1} \mathcal{G}) \cong Rf_! \mathcal{F} \otimes^L \mathcal{G}\).
Verdier duality: for \(a \colon X \to \mathrm{pt}\), the dualizing complex is \(\omega_X = a^! \ZZ\), and for a topological manifold of dimension \(n\) it is the orientation sheaf placed in degree \(-n\). For a closed inclusion \(i \colon Z \hookrightarrow X\), \(i^! \mathcal{F}\) is the derived functor of sections of \(\mathcal{F}\) supported in \(Z\), restricted to \(Z\).
The same formalism holds for constructible sheaves on complex algebraic varieties and for \(\ell\)-adic sheaves on schemes of finite type over a field. For a topological manifold \(X\) of dimension \(n\) with orientation sheaf \(\mathrm{or}_X\), Verdier duality for \(a\colon X\to\mathrm{pt}\) gives Poincaré duality \(H^i_c(X,\QQ)^\vee\cong H^{n-i}(X,\mathrm{or}_X\otimes\QQ)\).
Proper pushforward preserves constructible $\ell$-adic sheaves (Tag T-LADICPROPER)
Let \(X\) be a scheme of finite type over a field and \(\ell\) a prime invertible on \(X\). A constructible \(\ell\)-adic sheaf is a projective system \(\mcf = (\mcf_n)_{n \geq 1}\) of constructible étale sheaves of \(\ZZ/\ell^n\)-modules with \(\mcf_{n+1} \otimes_{\ZZ/\ell^{n+1}} \ZZ/\ell^n \cong \mcf_n\).
(Finiteness of proper pushforward)
Let \(f \colon X \to Y\) be a morphism of schemes of finite type over a field, with \(\ell\) invertible, and \(\mcf\) a constructible \(\ell\)-adic sheaf on \(X\). Then \(R^i f_! \mcf = (R^i f_! \mcf_n)_n\) is, up to Artin–Rees equivalence, a constructible \(\ell\)-adic sheaf on \(Y\).
For \(Y = \Spec \overline{\FF}_q\), this is the statement that \(H^i_c(X_{\overline{\FF}_q}, \QQ_\ell) = \big(\varprojlim_n H^i_c(X, \ZZ/\ell^n)\big) \otimes \QQ_\ell\) is finite-dimensional, which is what the Grothendieck–Lefschetz trace formula needs.